Regression with Panel Data

(SW Ch. 8)

A panel dataset contains observations on multiple

entities (individuals), where each entity is observed at

two or more points in time.

Examples:

Data on 420 California school districts in 1999 and

again in 2000, for 840 observations total.

Data on 50 U.S. states, each state is observed in 3

years, for a total of 150 observations.

Data on 1000 individuals, in four different months,

for 4000 observations total.8-1

Notation for panel data

A double subscript distinguishes entities (states) and

time periods (years)

i = entity (state), n = number of entities,

so i = 1,…,n

t = time period (year), T = number of time periods

so t =1,…,T

Data: Suppose we have 1 regressor. The data are:

(Xit, Yit), i = 1,…,n, t = 1,…,T

8-2

Panel data notation, ctd.

Panel data with k regressors:

(X1it, X2it,…,Xkit, Yit), i = 1,…,n, t = 1,…,T

n = number of entities (states)

T = number of time periods (years)

Some jargon…

Another term for panel data is longitudinal data

balanced panel: no missing observations

unbalanced panel: some entities (states) are not

observed for some time periods (years)

8-3

Why are panel data useful?

With panel data we can control for factors that:

Vary across entities (states) but do not vary over time

Could cause omitted variable bias if they are omitted

are unobserved or unmeasured – and therefore cannot

be included in the regression using multiple

regression

Here’s the key idea:

If an omitted variable does not change over time, then

any changes in Y over time cannot be caused by the

omitted variable.

8-4

Example of a panel data set:

Traffic deaths and alcohol taxes

Observational unit: a year in a U.S. state

48 U.S. states, so n = of entities = 48

7 years (1982,…, 1988), so T = # of time periods = 7

Balanced panel, so total # observations = 748 = 336

Variables:

Traffic fatality rate (# traffic deaths in that state in

that year, per 10,000 state residents)

Tax on a case of beer

Other (legal driving age, drunk driving laws, etc.)

8-5

Traffic death data for 1982

Higher alcohol taxes, more traffic deaths?8-6

Traffic death data for 1988

Higher alcohol taxes, more traffic deaths?

8-7

Why might there be higher more traffic deaths in states

that have higher alcohol taxes?

Other factors that determine traffic fatality rate:

Quality (age) of automobiles

Quality of roads

“Culture” around drinking and driving

Density of cars on the road

8-8

These omitted factors could cause omitted variable bias.

Example #1: traffic density. Suppose:

(i) High traffic density means more traffic deaths

(ii) (Western) states with lower traffic density have

lower alcohol taxes

Then the two conditions for omitted variable bias are

satisfied. Specifically, “high taxes” could reflect “high

traffic density” (so the OLS coefficient would be biased

positively – high taxes, more deaths)

Panel data lets us eliminate omitted variable bias when

the omitted variables are constant over time within a

given state.

8-9

Example #2: cultural attitudes towards drinking and

driving

(i) arguably are a determinant of traffic deaths; and

(ii) potentially are correlated with the beer tax, so beer

taxes could be picking up cultural differences

(omitted variable bias).

Then the two conditions for omitted variable bias are

satisfied. Specifically, “high taxes” could reflect

“cultural attitudes towards drinking” (so the OLS

coefficient would be biased)

Panel data lets us eliminate omitted variable bias when

the omitted variables are constant over time within a

given state.

8-10

Panel Data with Two Time Periods

(SW Section 8.2)

Consider the panel data model,

FatalityRateit = 0 + 1BeerTaxit + 2Zi + uit

Zi is a factor that does not change over time (density), at

least during the years on which we have data.

Suppose Zi is not observed, so its omission could

result in omitted variable bias.

The effect of Zi can be eliminated using T = 2 years.

8-11

The key idea:

Any change in the fatality rate from 1982 to 1988

cannot be caused by Zi, because Zi (by assumption)

does not change between 1982 and 1988.

The math: consider fatality rates in 1988 and 1982:

FatalityRatei1988 = 0 + 1BeerTaxi1988 + 2Zi + ui1988

FatalityRatei1982 = 0 + 1BeerTaxi1982 + 2Zi + ui1982

Suppose E(uit|BeerTaxit, Zi) = 0.

Subtracting 1988 – 1982 (that is, calculating the change),

eliminates the effect of Zi…

8-12

FatalityRatei1988 = 0 + 1BeerTaxi1988 + 2Zi + ui1988

FatalityRatei1982 = 0 + 1BeerTaxi1982 + 2Zi + ui1982

so

FatalityRatei1988 – FatalityRatei1982 =

1(BeerTaxi1988 – BeerTaxi1982) + (ui1988 – ui1982)

The new error term, (ui1988 – ui1982), is uncorrelated

with either BeerTaxi1988 or BeerTaxi1982.

This “difference” equation can be estimated by OLS,

even though Zi isn’t observed.

The omitted variable Zi doesn’t change, so it cannot

be a determinant of the change in Y

8-13

Example: Traffic deaths and beer taxes

1982 data:

= 2.01 + 0.15BeerTax (n = 48)

(.15) (.13)

1988 data:

= 1.86 + 0.44BeerTax (n = 48)

(.11) (.13)

Difference regression (n = 48)

= –.072 – 1.04(BeerTax1988–BeerTax1982)

(.065) (.36)

8-14

8-15

Fixed Effects Regression

(SW Section 8.3)

What if you have more than 2 time periods (T > 2)?

Yit = 0 + 1Xit + 2Zi + ui, i =1,…,n, T = 1,…,T

We can rewrite this in two useful ways:

1. “n-1 binary regressor” regression model

2. “Fixed Effects” regression model

We first rewrite this in “fixed effects” form. Suppose we

have n = 3 states: California, Texas, Massachusetts.

8-16

Yit = 0 + 1Xit + 2Zi + ui, i =1,…,n, T = 1,…,T

Population regression for California (that is, i = CA):

YCA,t = 0 + 1XCA,t + 2ZCA + uCA,t

= (0 + 2ZCA) + 1XCA,t + uCA,t

or

YCA,t = CA + 1XCA,t + uCA,t

CA = 0 + 2ZCA doesn’t change over time

CA is the intercept for CA, and 1 is the slope

The intercept is unique to CA, but the slope is the

same in all the states: parallel lines.

8-17

For TX:

YTX,t = 0 + 1XTX,t + 2ZTX + uTX,t

= (0 + 2ZTX) + 1XTX,t + uTX,t

or

YTX,t = TX + 1XTX,t + uTX,t, where TX = 0 + 2ZTX

Collecting the lines for all three states:

YCA,t = CA + 1XCA,t + uCA,t

YTX,t = TX + 1XTX,t + uTX,t

YMA,t = MA + 1XMA,t + uMA,t

or

Yit = i + 1Xit + uit, i = CA, TX, MA, T = 1,…,T

8-18

The regression lines for each state in a picture

Recall (Fig. 6.8a) that shifts in the intercept can be

represented using binary regressors…

Y = CA + 1X

Y = TX + 1X

Y = MA+ 1X

MA

TX

CA

Y

X

MA

TX

CA

8-19

In binary regressor form:

Yit = 0 + CADCAi + TXDTXi + 1Xit + uit

DCAi = 1 if state is CA, = 0 otherwise

DTXt = 1 if state is TX, = 0 otherwise

leave out DMAi (why?)

Y = CA + 1X

Y = TX + 1X

Y = MA+ 1X

MA

TX

CA

Y

X

MA

TX

CA

8-20

Summary: Two ways to write the fixed effects model

“n-1 binary regressor” form

Yit = 0 + 1Xit + 2D2i + … + nDni + ui

where D2i = , etc.

“Fixed effects” form:

Yit = 1Xit + i + ui

i is called a “state fixed effect” or “state effect” – it

is the constant (fixed) effect of being in state i

8-21

Fixed Effects Regression: Estimation

Three estimation methods:

1. “n-1 binary regressors” OLS regression

2. “Entity-demeaned” OLS regression

3. “Changes” specification (only works for T = 2)

These three methods produce identical estimates of the

regression coefficients, and identical standard errors.

We already did the “changes” specification (1988

minus 1982) – but this only works for T = 2 years

Methods #1 and #2 work for general T

Method #1 is only practical when n isn’t too big

8-22

1. “n-1 binary regressors” OLS regression

Yit = 0 + 1Xit + 2D2i + … + nDni + ui (1)

where D2i = etc.

First create the binary variables D2i,…,Dni

Then estimate (1) by OLS

Inference (hypothesis tests, confidence intervals) is as

usual (using heteroskedasticity-robust standard errors)

This is impractical when n is very large (for example if

n = 1000 workers)

8-23

2. “Entity-demeaned” OLS regression

The fixed effects regression model:

Yit = 1Xit + i + ui

The state averages satisfy:

= i + 1 +

Deviation from state averages:

Yit – = 1 +

8-24

Entity-demeaned OLS regression, ctd.

Yit – = 1 +

or

= 1 +

where = Yit – and = Xit –

For i=1 and t = 1982, is the difference between the

fatality rate in Alabama in 1982, and its average value

in Alabama averaged over all 7 years.

8-25

Entity-demeaned OLS regression, ctd.

= 1 + (2)

where = Yit – , etc.

First construct the demeaned variables and

Then estimate (2) by regressing on using OLS

Inference (hypothesis tests, confidence intervals) is as

usual (using heteroskedasticity-robust standard errors)

This is like the “changes” approach, but instead Yit is

deviated from the state average instead of Yi1.

This can be done in a single command in STATA

8-26

Example: Traffic deaths and beer taxes in STATA

. areg vfrall beertax, absorb(state) r;

Regression with robust standard errors Number of obs = 336 F( 1, 287) = 10.41 Prob > F = 0.0014 R-squared = 0.9050 Adj R-squared = 0.8891 Root MSE = .18986

------------------------------------------------------------------------------ | Robust vfrall | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- beertax | -.6558736 .2032797 -3.23 0.001 -1.055982 -.2557655 _cons | 2.377075 .1051515 22.61 0.000 2.170109 2.584041-------------+---------------------------------------------------------------- state | absorbed (48 categories)

“areg” automatically de-means the data

this is especially useful when n is large

the reported intercept is arbitrary

8-27

Example, ctd.

For n = 48, T = 7:

= –.66BeerTax + State fixed effects

(.20)

Should you report the intercept?

How many binary regressors would you include to

estimate this using the “binary regressor” method?

Compare slope, standard error to the estimate for the

1988 v. 1982 “changes” specification (T = 2, n = 48):

= –.072 – 1.04(BeerTax1988–BeerTax1982)

(.065) (.36)8-28

Regression with Time Fixed Effects

(SW Section 8.4)

An omitted variable might vary over time but not across

states:

Safer cars (air bags, etc.); changes in national laws

These produce intercepts that change over time

Let these changes (“safer cars”) be denoted by the

variable St, which changes over time but not states.

The resulting population regression model is:

Yit = 0 + 1Xit + 2Zi + 3St + uit

8-29

Time fixed effects only

Yit = 0 + 1Xit + 3St + uit

In effect, the intercept varies from one year to the next:

Yi,1982 = 0 + 1Xi,1982 + 3S1982 + ui,1982

= (0 + 3S1982) + 1Xi,1982 + ui,1982

or

Yi,1982 = 1982 + 1Xi,1982 + ui,1982, 1982 = 0 + 3S1982

Similarly,

Yi,1983 = 1983 + 1Xi,1983 + ui,1983, 1983 = 0 + 3S1983

etc.

8-30

Two formulations for time fixed effects

1. “Binary regressor” formulation:

Yit = 0 + 1Xit + 2B2t + … TBTt + uit

where B2t = , etc.

2. “Time effects” formulation:

Yit = 1Xit + t + uit

8-31

Time fixed effects: estimation methods

1. “T-1 binary regressors” OLS regression

Yit = 0 + 1Xit + 2B2it + … TBTit + uit

Create binary variables B2,…,BT

B2 = 1 if t = year #2, = 0 otherwise

Regress Y on X, B2,…,BT using OLS

Where’s B1?

2. “Year-demeaned” OLS regression

Deviate Yit, Xit from year (not state) averages

Estimate by OLS using “year-demeaned” data

8-32

State and Time Fixed Effects

Yit = 0 + 1Xit + 2Zi + 3St + uit

1. “Binary regressor” formulation:

Yit = 0 + 1Xit + 2D2i + … + nDni

+ 2B2t + … TBTt + uit

2. “State and time effects” formulation:

Yit = 1Xit + i + t + uit

8-33

State and time effects: estimation methods

1. “n-1 and T-1 binary regressors” OLS regression

Create binary variables D2,…,Dn

Create binary variables B2,…,BT

Regress Y on X, D2,…,Dn, B2,…,BT using OLS

What about D1 and B1?

2. “State- and year-demeaned” OLS regression

Deviate Yit, Xit from year and state averages

Estimate by OLS using “year- and state-

demeaned” data

These two methods can be combined too.

STATA example: Traffic deaths…. gen y83=(year==1983);

8-34

. gen y84=(year==1984);

. gen y85=(year==1985);

. gen y86=(year==1986);

. gen y87=(year==1987);

. gen y88=(year==1988);

. areg vfrall beertax y83 y84 y85 y86 y87 y88, absorb(state) r;

Regression with robust standard errors Number of obs = 336 F( 7, 281) = 3.70 Prob > F = 0.0008 R-squared = 0.9089 Adj R-squared = 0.8914 Root MSE = .18788------------------------------------------------------------------------------ | Robust vfrall | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- beertax | -.6399799 .2547149 -2.51 0.013 -1.141371 -.1385884 y83 | -.0799029 .0502708 -1.59 0.113 -.1788579 .0190522 y84 | -.0724206 .0452466 -1.60 0.111 -.161486 .0166448 y85 | -.1239763 .0460017 -2.70 0.007 -.214528 -.0334246 y86 | -.0378645 .0486527 -0.78 0.437 -.1336344 .0579055 y87 | -.0509021 .0516113 -0.99 0.325 -.1524958 .0506917 y88 | -.0518038 .05387 -0.96 0.337 -.1578438 .0542361 _cons | 2.42847 .1468565 16.54 0.000 2.139392 2.717549-------------+---------------------------------------------------------------- state | absorbed (48 categories)

Go to section for other ways to do this in STATA!

8-35

Some Theory: The Fixed Effects Regression

Assumptions (SW App. 8.2)

For a single X:

Yit = 1Xit + i + uit, i = 1,…,n, t = 1,…, T

1. E(uit|Xi1,…,XiT,i) = 0.

2. (Xi1,…,XiT,Yi1,…,YiT), i =1,…,n, are i.i.d. draws from

their joint distribution.

3. (Xit, uit) have finite fourth moments.

4. There is no perfect multicollinearity (multiple X’s)

5. corr(uit,uis|Xit,Xis,i) = 0 for t s.

Assumptions 3&4 are identical; 1, 2, differ; 5 is new

8-36

Assumption #1: E(uit|Xi1,…,XiT,i) = 0

uit has mean zero, given the state fixed effect and the

entire history of the X’s for that state

This is an extension of the previous multiple

regression Assumption #1

This means there are no omitted lagged effects (any

lagged effects of X must enter explicitly)

Also, there is not feedback from u to future X:

oWhether a state has a particularly high fatality rate

this year doesn’t subsequently affect whether it

increases the beer tax.

oWe’ll return to this when we take up time series

data.

8-37

Assumption #2: (Xi1,…,XiT,Yi1,…,YiT), i =1,…,n, are

i.i.d. draws from their joint distribution.

This is an extension of Assumption #2 for multiple

regression with cross-section data

This is satisfied if entities (states, individuals) are

randomly sampled from their population by simple

random sampling, then data for those entities are

collected over time.

This does not require observations to be i.i.d. over

time for the same entity – that would be unrealistic

(whether a state has a mandatory DWI sentencing law

this year is strongly related to whether it will have that

law next year).

8-38

Assumption #5: corr(uit,uis|Xit,Xis,i) = 0 for t s This is new.

This says that (given X), the error terms are

uncorrelated over time within a state.

For example, uCA,1982 and uCA,1983 are uncorrelated

Is this plausible? What enters the error term?

oEspecially snowy winter

oOpening major new divided highway

oFluctuations in traffic density from local economic

conditions

Assumption #5 requires these omitted factors entering

uit to be uncorrelated over time, within a state.

8-39

What if Assumption #5 fails: corr(uit,uis|Xit,Xis,i) 0?

A useful analogy is heteroskedasticity.

OLS panel data estimators of 1 are unbiased,

consistent

The OLS standard errors will be wrong – usually the

OLS standard errors understate the true uncertainty

Intuition: if uit is correlated over time, you don’t have

as much information (as much random variation) as you

would were uit uncorrelated.

This problem is solved by using “heteroskedasticity and

autocorrelation-consistent standard errors” – we return

to this when we focus on time series regression

Application: Drunk Driving Laws and Traffic Deaths

8-40

(SW Section 8.5)

Some facts

Approx. 40,000 traffic fatalities annually in the U.S.

1/3 of traffic fatalities involve a drinking driver

25% of drivers on the road between 1am and 3am

have been drinking (estimate)

A drunk driver is 13 times as likely to cause a fatal

crash as a non-drinking driver (estimate)

8-41

Drunk driving laws and traffic deaths, ctd.

Public policy issues

Drunk driving causes massive externalities (sober

drivers are killed, etc. etc.) – there is ample

justification for governmental intervention

Are there any effective ways to reduce drunk driving?

If so, what?

What are effects of specific laws:

omandatory punishment

ominimum legal drinking age

oeconomic interventions (alcohol taxes)

8-42

The drunk driving panel data set

n = 48 U.S. states, T = 7 years (1982,…,1988) (balanced)

Variables

Traffic fatality rate (deaths per 10,000 residents)

Tax on a case of beer (Beertax)

Minimum legal drinking age

Minimum sentencing laws for first DWI violation:

oMandatory Jail

oManditory Community Service

ootherwise, sentence will just be a monetary fine

Vehicle miles per driver (US DOT)

State economic data (real per capita income, etc.)8-43

Why might panel data help?

Potential OV bias from variables that vary across states

but are constant over time:

oculture of drinking and driving

oquality of roads

ovintage of autos on the road

use state fixed effects

Potential OV bias from variables that vary over time

but are constant across states:

oimprovements in auto safety over time

ochanging national attitudes towards drunk driving

use time fixed effects

8-44

8-45

8-46

Empirical Analysis: Main Results

Sign of beer tax coefficient changes when fixed state

effects are included

Fixed time effects are statistically significant but do not

have big impact on the estimated coefficients

Estimated effect of beer tax drops when other laws are

included as regressor

The only policy variable that seems to have an impact is

the tax on beer – not minimum drinking age, not

mandatory sentencing, etc.

The other economic variables have plausibly large

coefficients: more income, more driving, more deaths

8-47

Extensions of the “n-1 binary regressor” approach

The idea of using many binary indicators to eliminate

omitted variable bias can be extended to non-panel data –

the key is that the omitted variable is constant for a group

of observations, so that in effect it means that each group

has its own intercept.

Example: Class size problem.

Suppose funding and curricular issues are determined

at the county level, and each county has several

districts. Resulting omitted variable bias could be

addressed by including binary indicators, one for each

county (omit one to avoid perfect multicollinearity).

8-48

Summary: Regression with Panel Data

(SW Section 8.6)

Advantages and limitations of fixed effects regression

Advantages

You can control for unobserved variables that:

ovary across states but not over time, and/or

ovary over time but not across states

More observations give you more information

Estimation involves relatively straightforward

extensions of multiple regression

8-49

Fixed effects estimation can be done three ways:

1. “Changes” method when T = 2

2. “n-1 binary regressors” method when n is small

3. “Entity-demeaned” regression

Similar methods apply to regression with time fixed

effects and to both time and state fixed effects

Statistical inference: like multiple regression.

Limitations/challenges

Need variation in X over time within states

Time lag effects can be important

Standard errors might be too low (errors might be

correlated over time)

8-50